Graphics Programs Reference
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Central difference approximationsfor otherderivatives can beobtained from
Eqs. (a)-(h) in a similar manner. For example, eliminating f ( x )fromEqs. (f ) and (h)
and solving for f ( x ) yield
f ( x
+
2 h )
2 f ( x
+
h )
+
2 f ( x
h )
f ( x
2 h )
f ( x )
( h 2 )
=
+ O
(5.3)
2 h 3
The approximation
+
+
+
+
f ( x
2 h )
4 f ( x
h )
6 f ( x )
4 f ( x
h )
f ( x
2 h )
f (4) ( x )
( h 2 )
=
+ O
(5.4)
h 4
is available fromEq. (e) and (g) after eliminating f ( x ). Table 5.1 summarizes the
results.
+
+
f ( x
2 h )
f ( x
h )
f ( x )
f ( x
h )
f ( x
2 h )
2 hf ( x )
1
0
1
h 2 f ( x )
1
2
1
2 h 3 f ( x )
1
2
0
2
1
h 4 f (4) ( x )
1
4
6
4
1
Table 5.1. Coefficients of central finite difference approximations
of
O
( h 2 )
First Noncentral Finite Difference Approximations
Central finite difference approximations are not always usable.For example, consider
the situationwhere the functionis givenat the n discrete points x 1 ,
x 2 ,...,
x n .Since
central differences use values of the function on each sideof x , we wouldbe unable to
compute the derivatives at x 1 and x n .Clearly, there is aneed for finite difference
expressionsthat requireevaluations of the function only on onesideof x . These
expressions arecalled forward and backward finite difference approximations.
Noncentral finite differences can also beobtained fromEqs. (a)-(h).Solving
Eq. (a)for f ( x ) we get
h 2
6
h 3
4!
f ( x
+
h )
f ( x )
h
2 f ( x )
f ( x )
f ( x )
f (4) ( x )
=
−···
h
Keeping only the first term on the right-hand sideleads to the first forward difference
approximation
f ( x
+
h )
f ( x )
f ( x )
=
+ O
( h )
(5.5)
h
Similarly, Eq. (b) yields the first backward difference approximation
f ( x )
f ( x
h )
f ( x )
=
+ O
( h )
(5.6)
h
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